Splitting a Character's Body
When you split a smooth shaded object, the normals are recalculated. Their direction is depending on the vertex before and the one after each vertex. If you take the vertex "after" out of the equasion, the normals at the very edge will be modified. Hence the lighting will be treated differently exactly where the split has occured (which is what you are seeing).

The ppl. in the Unity forums were correct, however I am unaware of a method of splitting an object directly and keeping the normals at the split. 
Using multiple materials (can be the same material in multiple slots) does pretty much that, but internally, not visibly. 
So a crude approach could be to hide submeshes in Unity. 
Have not tried this, since most ppl. don't split naked characters, but rather hide the seams, but:
Split your object by material slots (assigning them to the faces in edit mode). 
In Unity use: 
myMesh.GetTriangles(mySubmeshIndex);
to show only the triangles that belong to the body part. 
There might be a better solution, but not one I can think of right now.